document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=483897>Question 483897</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the probability that a bullet fired from a point will hit the target is 1/3. Three such bullets are fired simultaneously  towards the target from that very point. What is the probability that the target will be hit? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #331141 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=331141\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> p = probability that the target will be hit = 1/3<BR>\n" );
document.write( "q = probability that the target will not be hit = 1 - 1/3 = 2/3<BR>\n" );
document.write( "p + q = 1, as it should, since either the target will be hit or the target will not be hit.\r<BR>\n" );
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document.write( "the probability that the target will be hit is equal to 1 - the probability that the target will not be hit.\r<BR>\n" );
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document.write( "the probability that the target will not be hit is 2/3 * 2/3 * 2/3 because you are firing at the target 3 times and each time the probability that the target will not be hit is 2/3.\r<BR>\n" );
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document.write( "the probability that the target will not be hit is therefore 8/27 (2*2*2)/(3*3*3).\r<BR>\n" );
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document.write( "since the total probability is 1 which is equivalent to 27/27, then the probability that the target will be hit is equal to 27/27 - 8/27 = 19/27\r<BR>\n" );
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document.write( "this is the short way to solve this problem.\r<BR>\n" );
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document.write( "the long way is to look at the probability that:\r<BR>\n" );
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document.write( "a.  the target is hit 1 time only.<BR>\n" );
document.write( "b.  the target is hit 2 times only.<BR>\n" );
document.write( "c.  the target is hit 3 times only.\r<BR>\n" );
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document.write( "since the target can't be hit any other way, then the probability that the target will be hit will be the sum of these 3 independent events.\r<BR>\n" );
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document.write( "the probability that any of the posssible events is given by the formula:\r<BR>\n" );
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document.write( "nCx * p^x * q^(n-x)\r<BR>\n" );
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document.write( "n is the total number of shots fired at the target.<BR>\n" );
document.write( "x is the number of times the target can be hit exactly.\r<BR>\n" );
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document.write( "this comes out to be:\r<BR>\n" );
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document.write( "3C0 * p^0 * q^3<BR>\n" );
document.write( "3C1 * p^1 * q^2<BR>\n" );
document.write( "3C2 * p^2 * q^1<BR>\n" );
document.write( "3C3 * p^3 * q^0\r<BR>\n" );
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document.write( "We already solved for 3C0 * p^0 * q^3<BR>\n" );
document.write( "that number came out to be 8/27<BR>\n" );
document.write( "3C0 equals 3!/(0!*3!) which equals 1<BR>\n" );
document.write( "p^0 = 1<BR>\n" );
document.write( "q^3 = (2/3) * (2/3) * (2/3) which equals 8/27<BR>\n" );
document.write( "3C0 * p^0 * q^3 = 1 * 1 * 8/27 = 8/27\r<BR>\n" );
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document.write( "We can solve for 3C3 * p^3 * q^0 easy enough.<BR>\n" );
document.write( "3C3 equals (3! / (3!*0!) which equals 1<BR>\n" );
document.write( "p^3 = (1/3) * (1/3) * (1/3) which equals 1/27<BR>\n" );
document.write( "q^0 = 1<BR>\n" );
document.write( "3C3 * p^3 * q^0 = 1 * 1/27 * 1 = 1/27\r<BR>\n" );
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document.write( "We can solve for 3C1 * p^1 * q^2 as follows:<BR>\n" );
document.write( "3C1 = 3! / (1!*2!) = 3!/2 = 6/2 = 3<BR>\n" );
document.write( "p^1 = 1/3<BR>\n" );
document.write( "q^2 = 2/3 * 2/3 = 4/9<BR>\n" );
document.write( "3C1 * p^1 * q^2 = 3 * 1/3 * 4/9 = 3/3 * 4/9 = 12/27\r<BR>\n" );
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document.write( "We can solve for 3C2 * p^2 * q^1 as follows:<BR>\n" );
document.write( "3C2 = 3! / (2!*1!) = 3!/2 = 6/2 = 3<BR>\n" );
document.write( "p^2 = 1/3 * 1/3 = 1/9<BR>\n" );
document.write( "q^1 = 2/3<BR>\n" );
document.write( "3C2 * p^2 * q^1 = 3 * 1/9 * 2/3 = 3/9 * 2/3 = 6/27\r<BR>\n" );
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document.write( "our total probabilities are:<BR>\n" );
document.write( "p(0) = 8/27<BR>\n" );
document.write( "p(1) = 12/27<BR>\n" );
document.write( "p(2) =  6/27<BR>\n" );
document.write( "p(3) =  1/27\r<BR>\n" );
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document.write( "The total probability is 8 + 12 + 6 + 1 = 27/27 = 1 as it should.\r<BR>\n" );
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document.write( "The probability we will hit the target is p(1) + p(2) + p(3) = 12 + 6 + 1 = 19/27.\r<BR>\n" );
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document.write( "That's the same number we got before when we did 1 - p(0).\r<BR>\n" );
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document.write( "We're good and that's your answer.\r<BR>\n" );
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document.write( "The probability that the target will be hit is 19/27.\r<BR>\n" );
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document.write( "nCx is the combination formula and it is defined as n! / (x! * (n-x)!)\r<BR>\n" );
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document.write( "You need to multiply the probability of each occurrence by the number of ways that occurrence can occur.\r<BR>\n" );
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document.write( "An example:\r<BR>\n" );
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document.write( "You have 3 shots.\r<BR>\n" );
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document.write( "You want to get 1 out of 3 shots.\r<BR>\n" );
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document.write( "The probability is 3C1 * p(1) * q(2).\r<BR>\n" );
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document.write( "Assume h means hit and x means miss.\r<BR>\n" );
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document.write( "The number of ways is 3C2 which is equal to 3 (we did that above).\r<BR>\n" );
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document.write( "Those possible ways are:\r<BR>\n" );
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document.write( "hmm<BR>\n" );
document.write( "mhm<BR>\n" );
document.write( "mmh\r<BR>\n" );
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document.write( "h =  hit<BR>\n" );
document.write( "m = miss\r<BR>\n" );
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document.write( "The probability of each of these ways is the same.\r<BR>\n" );
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document.write( "The probability of getting 1 out of 3 is p(hmm) + (pmhm) + p(mmh).\r<BR>\n" );
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document.write( "Since the probability of hmm is equal to the probability of mhm is equal to the probability of mmh, then your probability of getting 1 out of 3 can be expressed as 3 * p(hmm).\r<BR>\n" );
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document.write( " </SPAN>\n" );
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